Quasisymmetric boundary correspondence and Grtzsch problem

Two necessary and sufficient conditions for the validity of the conjecture K 0(h)=K 1(h) are given, which are independent of the complex dilatations of extremal quasiconformal mappings, where K 0(h) is the maximal conformal modulus dilatation of the boundary homeomorphism h, K 1(h) is the maximal dilatation of extremal quasiconformal mappings that agree with h on the boundary. In addition, when the complex dilatation of an extremal quasiconformal mapping is known, the proof of the result simplifies Reich and Chen Jixiu-Chen Zhiguo’s result.

A unified approach to the weighted Grtzsch and Nitsche problems for mappings of finite distortion

This note deals with the existence and uniqueness of a minimiser of the following Grtzsch-type problem inf f ∈F∫∫_(Q_1)φ(K(z,f))λ(x)dxdyunder some mild conditions,where F denotes the set of all homeomorphims f with finite linear distortion K(z,f)between two rectangles Q_1 and Q_2 taking vertices into vertices,φ is a positive,increasing and convex function,and λ is a positive weight function.A similar problem of Nitsche-type,which concerns the minimiser of some weighted functional for mappings between two annuli,is also discussed.As by-products,our discussion gives a unified approach to some known results in the literature concerning the weighted Grtzsch and Nitsche problems.

广义Grtzsch环函数的几个性质(英文)

设0<a,r<1,r′=1-r2,Ka(r)为第一类广义椭圆积分,定义广义Gr¨otzsch环函数为μa(r)={π/[2sin(πa)]}Ka(r′)/Ka(r),μa(r)出现于广义Ramanujan模方程,而μ1/2(r)即为拟共形理论中平面Gr¨otzsch环B2\[0,r]的共形模。本文揭示了μa(r)和广义椭圆积分的若干性质,这些结果将被用于研究广义Ramanujan模方程及其解的性态。

关于拟共形映照Mori常数的注记

讨论拟共形映照的偏差问题。